Suppose a consumer's preferences for goods X and Y can be described by the utility function U(X, Y) = (AXR + BYR)1/R,
where A and B are positive numbers and R is a number that can be either positive or negative.a. Do the consumer's preferences satisfy the More-Is-Better Principle?
b. What is the consumer's marginal rate of substitution at the bundle (X, Y)? (The answer will be a formula that gives the MRS as a function of the consumption levels X and Y, and may also depend on the number A, B, and R.)
c. Do these preferences satisfy the declining MRS property? (Does your answer depend on the value of R?)
d. Show that, for R = 1, we have the case of perfect substitutes.
e. Show that as R approaches 0, this utility function is associated with the same preferences as the Cobb-Douglas utility function, U(X, Y) = XAYB.
f. Show that as R approaches -∞ we have the case of perfect complements.